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Chapter 13: Problem 12

Find the first partial derivatives of the following functions. $$f(x, y)=\ln (x / y)$$

Short Answer

Expert verified
Answer: The first partial derivatives are \(\frac{\partial f}{\partial x} = \frac{1}{x}\) and \(\frac{\partial f}{\partial y} = -\frac{1}{y}\).

Step by step solution

01

Rewrite the function using logarithm properties

We can rewrite the function using the property of logarithms: \(\ln(a / b) = \ln(a) - \ln(b)\). So, we can rewrite the given function as $$f(x, y) = \ln(x) - \ln(y).$$
02

Find the partial derivative with respect to x

We can now find the partial derivative of \(f(x, y)\) with respect to \(x\), denoted as \(\frac{\partial f}{\partial x}\). Using the properties of logarithm, we have: $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(\ln(x) - \ln(y)).$$ Since \(\ln(y)\) is a constant with respect to \(x\), its derivative will be zero. Therefore: $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x}(\ln(x)) - 0$$ The derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\). So, we have: $$\frac{\partial f}{\partial x} = \frac{1}{x}.$$
03

Find the partial derivative with respect to y

Now, we find the partial derivative of \(f(x, y)\) with respect to \(y\), denoted as \(\frac{\partial f}{\partial y}\). Using the properties of logarithm, we have: $$\frac{\partial f}{\partial y} = \frac{\partial}{\partial y}(\ln(x) - \ln(y)).$$ Since \(\ln(x)\) is a constant with respect to \(y\), its derivative will be zero. Therefore: $$\frac{\partial f}{\partial y} = 0 - \frac{\partial}{\partial y}(\ln(y))$$ The derivative of \(\ln(y)\) with respect to \(y\) is \(\frac{1}{y}\). So, we have: $$\frac{\partial f}{\partial y} = -\frac{1}{y}.$$
04

Write down the first partial derivatives

We've found the first partial derivatives of the given function, \(f(x, y) = \ln(x / y)\). The first partial derivatives are: $$\frac{\partial f}{\partial x} = \frac{1}{x}$$ and $$\frac{\partial f}{\partial y} = -\frac{1}{y}.$$

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Most popular questions from this chapter

Given three distinct noncollinear points \(A, B,\) and \(C\) in the plane, find the point \(P\) in the plane such that the sum of the distances \(|A P|+|B P|+|C P|\) is a minimum. Here is how to proceed with three points, assuming that the triangle formed by the three points has no angle greater than \(2 \pi / 3\left(120^{\circ}\right)\). a. Assume the coordinates of the three given points are \(A\left(x_{1}, y_{1}\right)\) \(B\left(x_{2}, y_{2}\right),\) and \(C\left(x_{3}, y_{3}\right) .\) Let \(d_{1}(x, y)\) be the distance between \(A\left(x_{1}, y_{1}\right)\) and a variable point \(P(x, y) .\) Compute the gradient of \(d_{1}\) and show that it is a unit vector pointing along the line between the two points. b. Define \(d_{2}\) and \(d_{3}\) in a similar way and show that \(\nabla d_{2}\) and \(\nabla d_{3}\) are also unit vectors in the direction of the line between the two points. c. The goal is to minimize \(f(x, y)=d_{1}+d_{2}+d_{3}\) Show that the condition \(f_{x}=f_{y}=0\) implies that \(\nabla d_{1}+\nabla d_{2}+\nabla d_{3}=0\). d. Explain why part (c) implies that the optimal point \(P\) has the property that the three line segments \(A P, B P,\) and \(C P\) all intersect symmetrically in angles of \(2 \pi / 3\). e. What is the optimal solution if one of the angles in the triangle is greater than \(2 \pi / 3\) (just draw a picture)? f. Estimate the Steiner point for the three points (0,0),(0,1) and (2,0)

The angle between two planes is the angle \(\theta\) between the normal vectors of the planes, where the directions of the normal vectors are chosen so that \(0 \leq \theta<\pi\) Find the angle between the planes \(5 x+2 y-z=0\) and \(-3 x+y+2 z=0\)

Ideal Gas Law Many gases can be modeled by the Ideal Gas Law, \(P V=n R T,\) which relates the temperature \((T,\) measured in Kelvin (K)), pressure ( \(P\), measured in Pascals (Pa)), and volume ( \(V\), measured in \(\mathrm{m}^{3}\) ) of a gas. Assume that the quantity of gas in question is \(n=1\) mole (mol). The gas constant has a value of \(R=8.3 \mathrm{m}^{3} \cdot \mathrm{Pa} / \mathrm{mol} \cdot \mathrm{K}.\) a. Consider \(T\) to be the dependent variable and plot several level curves (called isotherms) of the temperature surface in the region \(0 \leq P \leq 100,000\) and \(0 \leq V \leq 0.5.\) b. Consider \(P\) to be the dependent variable and plot several level curves (called isobars) of the pressure surface in the region \(0 \leq T \leq 900\) and \(0< V \leq 0.5.\) c. Consider \(V\) to be the dependent variable and plot several level curves of the volume surface in the region \(0 \leq T \leq 900\) and \(0 < P \leq 100,000.\)

Identify and briefly describe the surfaces defined by the following equations. $$x^{2}+4 y^{2}=1$$

a. Show that the point in the plane \(a x+b y+c z=d\) nearest the origin is \(P\left(a d / D^{2}, b d / D^{2}, c d / D^{2}\right),\) where \(D^{2}=a^{2}+b^{2}+c^{2} .\) Conclude that the least distance from the plane to the origin is \(|d| / D\). (Hint: The least distance is along a normal to the plane.) b. Show that the least distance from the point \(P_{0}\left(x_{0}, y_{0}, z_{0}\right)\) to the plane \(a x+b y+c z=d\) is \(\left|a x_{0}+b y_{0}+c z_{0}-d\right| / D\).
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